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2021 AMC 12B problems

All 25 problems from the 2021 AMC 12B, with answer choices, worked solutions and hints. Problems are roughly ordered by difficulty: 1–10 are the most approachable, 19–25 the hardest.

Problems

  1. 1Problem 1How many integer values of x satisfy |x| < 3π?Algebra
  2. 2Problem 2At a math contest, 57 students are wearing blue shirts, and another 75 students are wearing yellow shirts. The 132 students are assigned into 66…Number Theory
  3. 3Problem 3Suppose 2+cfrac 11+cfrac 12+cfrac 23+x=144/53. What is the value of x?Algebra
  4. 4Problem 4Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is 84, and the afternoon class’s mean score…Algebra
  5. 5Problem 5The point P(a,b) in the xy-plane is first rotated counterclockwise by 90° around the point (1,5) and then reflected about the line y=-x. The image of…Geometry
  6. 6Problem 6An inverted cone with base radius 12 cm and height 18 cm is full of water. The water is poured into a tall cylinder whose horizontal base has a…Geometry
  7. 7Problem 7Let N=34· 34· 63· 270. What is the ratio of the sum of the odd divisors of N to the sum of the even divisors of N?Number Theory
  8. 8Problem 8Three equally spaced parallel lines intersect a circle, creating three chords of lengths 38, 38, and 34. What is the distance between two adjacent…Geometry
  9. 9Problem 9What is the value of the following expression? dfrac log _2 80log _402-dfrac log _2 160log _202Algebra
  10. 10Problem 10Two distinct numbers are selected from the set {1,2,3,4,…,36,37} so that the sum of the remaining 35 numbers is the product of these two numbers…Algebra
  11. 11Problem 11Triangle ABC has AB=13, BC=14, and AC=15. Let P be the point on AC such that PC=10. There are exactly two points D and E on line BP such that…Geometry
  12. 12Problem 12Suppose that S is a finite set of positive integers. If the greatest integer in S is removed from S, then the average value (arithmetic mean) of the…Algebra
  13. 13Problem 13How many values of θ in the interval 0<θ≤ 2π satisfy the following equation? 1-3sin θ+5cos 3θ=0Geometry
  14. 14Problem 14Let ABCD be a rectangle and let DM be a segment perpendicular to the plane of ABCD. Suppose that DM has integer length, and the lengths of MA, MC…Geometry
  15. 15Problem 15The figure is constructed from 11 line segments, each of which has length 2. The area of pentagon ABCDE can be written as sqrt m+sqrt n, where m and…Geometry
  16. 16Problem 16Let g(x) be a polynomial with leading coefficient 1, whose three roots are the reciprocals of the three roots of f(x)=x^3+ax^2+bx+c, where 1< a< b<…Algebra
  17. 17Problem 17Let ABCD be an isosceles trapezoid having parallel bases AB and CD with AB> CD. Line segments from a point inside ABCD to the vertices divide the…Geometry
  18. 18Problem 18Let z be a complex number satisfying 12|z|^2 =2|z+2|^2+|z^2+1|^2+31. What is the value of z+6/z?Algebra
  19. 19Problem 19Two fair dice, each with at least 6 faces are rolled. On each face of each die is printed a distinct integer from 1 to the number of faces on that…Algebra
  20. 20Problem 20Let Q(z) and R(z) be the unique polynomials such that z^2021+1 = (z^2+z+1)Q(z)+R(z) and the degree of R is less than 2. What is R(z)?Algebra
  21. 21Problem 21Let S be the sum of all positive real numbers x for which x^2^sqrt 2=sqrt 2^ 2^x. Which of the following statements is true?Algebra
  22. 22Problem 22Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one “wall” among a set of several walls of bricks…Counting & Probability
  23. 23Problem 23Three balls are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is…Algebra
  24. 24Problem 24Let ABCD be a parallelogram with area 15. Points P and Q are the projections of A and C, respectively, onto the line BD; and points R and S are the…Geometry
  25. 25Problem 25Let S be the set of lattice points in the coordinate plane, both of whose coordinates are integers between 1 and 30, inclusive. Exactly 300 points in…Algebra

Practise the same ideas across every year on the topic pages, or browse the full AMC 12 archive.